Properties

Label 1764.3.z.n
Level $1764$
Weight $3$
Character orbit 1764.z
Analytic conductor $48.066$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,3,Mod(325,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.325");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1764.z (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(48.0655186332\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 12x^{14} + 114x^{12} - 336x^{10} + 755x^{8} - 336x^{6} + 114x^{4} - 12x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{6}\cdot 7^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{12} - \beta_{4}) q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{12} - \beta_{4}) q^{5} + \beta_{11} q^{11} + \beta_{13} q^{13} - 3 \beta_{4} q^{17} + (2 \beta_{13} - 2 \beta_{9} + \beta_{3}) q^{19} + ( - \beta_{11} + \beta_{10} + \beta_{5} + \beta_1) q^{23} + ( - \beta_{8} + 3 \beta_{2} + 3) q^{25} + ( - \beta_{10} + \beta_{5}) q^{29} + ( - 4 \beta_{9} - 5 \beta_{7}) q^{31} + (3 \beta_{8} - 3 \beta_{6} + 16 \beta_{2}) q^{37} + (\beta_{15} + \beta_{14} - \beta_{12} + \beta_{4}) q^{41} + ( - 2 \beta_{6} + 22) q^{43} + ( - \beta_{15} - 2 \beta_{12} - 2 \beta_{4}) q^{47} - 2 \beta_{11} q^{53} + (18 \beta_{13} - \beta_{7} + \beta_{3}) q^{55} + (\beta_{14} + 3 \beta_{4}) q^{59} + (13 \beta_{13} - 13 \beta_{9} + 12 \beta_{3}) q^{61} + (\beta_{11} - \beta_{5}) q^{65} + ( - 6 \beta_{8} + 32 \beta_{2} + 32) q^{67} + (2 \beta_{10} + \beta_{5}) q^{71} + ( - 21 \beta_{9} - 22 \beta_{7}) q^{73} + (8 \beta_{8} - 8 \beta_{6} + 54 \beta_{2}) q^{79} + ( - 2 \beta_{15} - 2 \beta_{14} - 2 \beta_{12} - 2 \beta_{4}) q^{83} + ( - 3 \beta_{6} + 84) q^{85} + (2 \beta_{15} + 9 \beta_{12} + 9 \beta_{4}) q^{89} + (2 \beta_{11} + \beta_1) q^{95} + (27 \beta_{13} + 2 \beta_{7} - 2 \beta_{3}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 24 q^{25} - 128 q^{37} + 352 q^{43} + 256 q^{67} - 432 q^{79} + 1344 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 12x^{14} + 114x^{12} - 336x^{10} + 755x^{8} - 336x^{6} + 114x^{4} - 12x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 473 \nu^{14} - 96474 \nu^{12} + 1132373 \nu^{10} - 10999257 \nu^{8} + 29348105 \nu^{6} - 62719633 \nu^{4} + 4563258 \nu^{2} - 481239 ) / 157522 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 8724 \nu^{14} + 103818 \nu^{12} - 984456 \nu^{10} + 2836231 \nu^{8} - 6333624 \nu^{6} + 2379102 \nu^{4} - 955212 \nu^{2} + 21780 ) / 78761 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 7388 \nu^{15} + 75216 \nu^{13} - 686249 \nu^{11} + 1014272 \nu^{9} - 1669588 \nu^{7} - 5856677 \nu^{5} - 234744 \nu^{3} + 24592 \nu ) / 78761 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 32730 \nu^{15} + 379529 \nu^{13} - 3575207 \nu^{11} + 9517884 \nu^{9} - 20532779 \nu^{7} + 1479396 \nu^{5} - 156013 \nu^{3} - 3968033 \nu ) / 315044 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 125820 \nu^{14} - 1459483 \nu^{12} + 13743738 \nu^{10} - 36588456 \nu^{8} + 78586038 \nu^{6} - 5687064 \nu^{4} + 599742 \nu^{2} + 3033171 ) / 157522 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 210\nu^{14} - 2436\nu^{12} + 22939\nu^{10} - 61068\nu^{8} + 131117\nu^{6} - 9492\nu^{4} + 1001\nu^{2} + 2310 ) / 226 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 34350 \nu^{15} + 398539 \nu^{13} - 3752165 \nu^{11} + 9988980 \nu^{9} - 21394552 \nu^{7} + 1552620 \nu^{5} - 163735 \nu^{3} + 92422 \nu ) / 78761 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 191009 \nu^{14} + 2326128 \nu^{12} - 22169357 \nu^{10} + 67895142 \nu^{8} - 154104811 \nu^{6} + 85598856 \nu^{4} - 23312730 \nu^{2} + 2454270 ) / 157522 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 186810 \nu^{15} + 2166991 \nu^{13} - 20405879 \nu^{11} + 54324348 \nu^{9} - 116648235 \nu^{7} + 8443812 \nu^{5} - 890461 \nu^{3} - 4213215 \nu ) / 315044 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 274170 \nu^{14} - 3181229 \nu^{12} + 29948503 \nu^{10} - 79728636 \nu^{8} + 170614699 \nu^{6} - 12392484 \nu^{4} + 1306877 \nu^{2} - 206391 ) / 157522 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 346926 \nu^{14} + 4174002 \nu^{12} - 39675882 \nu^{10} + 117756687 \nu^{8} - 265095942 \nu^{6} + 123410351 \nu^{4} - 40041792 \nu^{2} + \cdots + 4215021 ) / 157522 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 11593 \nu^{15} - 137995 \nu^{13} + 1308540 \nu^{11} - 3771965 \nu^{9} + 8418660 \nu^{7} - 3162305 \nu^{5} + 1198319 \nu^{3} - 28950 \nu ) / 7684 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 39393 \nu^{15} + 469469 \nu^{13} - 4451748 \nu^{11} + 12864843 \nu^{9} - 28640892 \nu^{7} + 10758391 \nu^{5} - 3342383 \nu^{3} + 98490 \nu ) / 18532 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3450630 \nu^{15} + 40028431 \nu^{13} - 376923817 \nu^{11} + 1003443204 \nu^{9} - 2153783649 \nu^{7} + 155968476 \nu^{5} - 16448003 \nu^{3} + \cdots - 52231323 \nu ) / 315044 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 39787 \nu^{15} + 481194 \nu^{13} - 4579212 \nu^{11} + 13778057 \nu^{9} - 31129685 \nu^{7} + 15713328 \nu^{5} - 4705218 \nu^{3} + 495319 \nu ) / 1921 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{14} + 28\beta_{9} - 7\beta_{7} - 25\beta_{4} ) / 98 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 11\beta_{11} + \beta_{10} - 7\beta_{8} + 7\beta_{6} - 11\beta_{5} - 147\beta_{2} + \beta_1 ) / 49 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -13\beta_{15} - 13\beta_{14} + 280\beta_{13} + 116\beta_{12} - 21\beta_{7} - 13\beta_{4} + 21\beta_{3} ) / 98 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 14\beta_{11} - 12\beta_{8} - 147\beta_{2} - 147 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -125\beta_{15} + 2464\beta_{13} + 844\beta_{12} - 2464\beta_{9} + 844\beta_{4} - 21\beta_{3} ) / 98 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 29\beta_{10} - 791\beta_{6} + 857\beta_{5} - 8379 ) / 49 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1121\beta_{14} - 21532\beta_{9} - 791\beta_{7} + 8033\beta_{4} ) / 98 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -1068\beta_{11} + 48\beta_{10} + 1008\beta_{8} - 1008\beta_{6} + 1068\beta_{5} + 10199\beta_{2} + 48\beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 9857 \beta_{15} + 9857 \beta_{14} - 187796 \beta_{13} - 58992 \beta_{12} - 8617 \beta_{7} + 9857 \beta_{4} + 8617 \beta_{3} ) / 98 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -65167\beta_{11} + 61943\beta_{8} + 617547\beta_{2} + 3163\beta _1 + 617547 ) / 49 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 86141\beta_{15} - 1636880\beta_{13} - 510556\beta_{12} + 1636880\beta_{9} - 510556\beta_{4} + 79947\beta_{3} ) / 98 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( -4032\beta_{10} + 77292\beta_{6} - 81130\beta_{5} + 766899 ) / 7 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( -751309\beta_{14} + 14264600\beta_{9} + 710325\beta_{7} - 5190321\beta_{4} ) / 98 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 4948765 \beta_{11} - 247801 \beta_{10} - 4718119 \beta_{8} + 4718119 \beta_{6} - 4948765 \beta_{5} - 46741443 \beta_{2} - 247801 \beta_1 ) / 49 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 6548641 \beta_{15} - 6548641 \beta_{14} + 124300820 \beta_{13} + 38652456 \beta_{12} + 6228103 \beta_{7} - 6548641 \beta_{4} - 6228103 \beta_{3} ) / 98 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{2}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
325.1
−1.45341 0.839125i
−0.516029 0.297929i
−2.55641 1.47595i
−0.293380 0.169383i
2.55641 + 1.47595i
0.293380 + 0.169383i
1.45341 + 0.839125i
0.516029 + 0.297929i
−1.45341 + 0.839125i
−0.516029 + 0.297929i
−2.55641 + 1.47595i
−0.293380 + 0.169383i
2.55641 1.47595i
0.293380 0.169383i
1.45341 0.839125i
0.516029 0.297929i
0 0 0 −5.33147 + 3.07813i 0 0 0 0 0
325.2 0 0 0 −5.33147 + 3.07813i 0 0 0 0 0
325.3 0 0 0 −3.68448 + 2.12723i 0 0 0 0 0
325.4 0 0 0 −3.68448 + 2.12723i 0 0 0 0 0
325.5 0 0 0 3.68448 2.12723i 0 0 0 0 0
325.6 0 0 0 3.68448 2.12723i 0 0 0 0 0
325.7 0 0 0 5.33147 3.07813i 0 0 0 0 0
325.8 0 0 0 5.33147 3.07813i 0 0 0 0 0
901.1 0 0 0 −5.33147 3.07813i 0 0 0 0 0
901.2 0 0 0 −5.33147 3.07813i 0 0 0 0 0
901.3 0 0 0 −3.68448 2.12723i 0 0 0 0 0
901.4 0 0 0 −3.68448 2.12723i 0 0 0 0 0
901.5 0 0 0 3.68448 + 2.12723i 0 0 0 0 0
901.6 0 0 0 3.68448 + 2.12723i 0 0 0 0 0
901.7 0 0 0 5.33147 + 3.07813i 0 0 0 0 0
901.8 0 0 0 5.33147 + 3.07813i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 325.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.3.z.n 16
3.b odd 2 1 inner 1764.3.z.n 16
7.b odd 2 1 inner 1764.3.z.n 16
7.c even 3 1 1764.3.d.g 8
7.c even 3 1 inner 1764.3.z.n 16
7.d odd 6 1 1764.3.d.g 8
7.d odd 6 1 inner 1764.3.z.n 16
21.c even 2 1 inner 1764.3.z.n 16
21.g even 6 1 1764.3.d.g 8
21.g even 6 1 inner 1764.3.z.n 16
21.h odd 6 1 1764.3.d.g 8
21.h odd 6 1 inner 1764.3.z.n 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.3.d.g 8 7.c even 3 1
1764.3.d.g 8 7.d odd 6 1
1764.3.d.g 8 21.g even 6 1
1764.3.d.g 8 21.h odd 6 1
1764.3.z.n 16 1.a even 1 1 trivial
1764.3.z.n 16 3.b odd 2 1 inner
1764.3.z.n 16 7.b odd 2 1 inner
1764.3.z.n 16 7.c even 3 1 inner
1764.3.z.n 16 7.d odd 6 1 inner
1764.3.z.n 16 21.c even 2 1 inner
1764.3.z.n 16 21.g even 6 1 inner
1764.3.z.n 16 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{8} - 56T_{5}^{6} + 2450T_{5}^{4} - 38416T_{5}^{2} + 470596 \) Copy content Toggle raw display
\( T_{11}^{8} + 364T_{11}^{6} + 131124T_{11}^{4} + 499408T_{11}^{2} + 1882384 \) Copy content Toggle raw display
\( T_{13}^{4} + 20T_{13}^{2} + 2 \) Copy content Toggle raw display
\( T_{19}^{8} - 136T_{19}^{6} + 14264T_{19}^{4} - 575552T_{19}^{2} + 17909824 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 56 T^{6} + 2450 T^{4} + \cdots + 470596)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + 364 T^{6} + 131124 T^{4} + \cdots + 1882384)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 20 T^{2} + 2)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} - 504 T^{6} + 198450 T^{4} + \cdots + 3087580356)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 136 T^{6} + 14264 T^{4} + \cdots + 17909824)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 1652 T^{6} + \cdots + 157218594064)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 2100 T^{2} + 725788)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} - 1160 T^{6} + \cdots + 49778964544)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 32 T^{3} + 1650 T^{2} + \cdots + 391876)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 4928 T^{2} + 3655694)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 44 T + 92)^{8} \) Copy content Toggle raw display
$47$ \( (T^{8} - 4928 T^{6} + \cdots + 36741733758016)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 1456 T^{6} + 2097984 T^{4} + \cdots + 481890304)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 5152 T^{6} + \cdots + 6953684616256)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 10388 T^{6} + \cdots + 654840832460164)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 64 T^{3} + 6600 T^{2} + \cdots + 6270016)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 5964 T^{2} + 1318492)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} - 24484 T^{6} + \cdots + 15\!\cdots\!84)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 108 T^{3} + 15020 T^{2} + \cdots + 11262736)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 19264 T^{2} + 86940896)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} - 22792 T^{6} + \cdots + 10\!\cdots\!56)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 14308 T^{2} + 235298)^{4} \) Copy content Toggle raw display
show more
show less